A circle has a radius of $5$. An arc in this circle has a central angle of $216^\circ$. What is the length of the arc? ${10\pi}$ ${216^\circ}$ $\color{#DF0030}{6\pi}$ ${5}$
Solution: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (5) = 10\pi$ The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{216^\circ}{360^\circ} = \dfrac{s}{10\pi}$ $\dfrac{3}{5} = \dfrac{s}{10\pi}$ $\dfrac{3}{5} \times 10\pi = s$ $6\pi = s$